A classical question ([1]}) raised by Stein is what are the relations between the Bergman kernel \(K\)  and the Szegö kernel \(S\)? On any bounded domain in \(\mathbb {C}\)  with Lipschitz boundary (see [2]}, [3]} for the \(C^\infty \)  smooth case and Proposition REF  for the Lipschitz case),
the Szegö kernel \(S\)  and the analytic capacity \(c_B\)  satisfy the identity \(c_B(z) = 2\pi S(z)\) . Using this and (REF ), we deduce the following inequality chain, which gives a relation between the on-diagonal Bergman and Szegö kernels:
\( \sqrt{ \pi K(z)} \ge c_{\beta } (z) \ge 2\pi S(z) \ge \sqrt{\frac{\pi }{v(\Omega )}} \ge \frac{2\pi }{\sigma (\partial \Omega )}.\) 
